Basically the top 4 in the premiership qualify for the champions league which is simple enough.
For the Europa League we have 3 spots. So the fir spot you get from finishing 5th, the second from winning the F.A. cup and the 3rd from winning the league cup. However you can only be in one European competition. So if anyone form the top 5 wins the F.A. Cup or League cup the Europa League spot then goes to the teams that finished 6th and if need be 7th as well.
You can also get demoted into the Europa League form the champions league if you finish 3rd in your champions league group, which is how Man Utd got into it.
All in all its a farce of a competition with a stupid format.